research article mesons in asymmetric hot and dense ...out of two mesons, four-quark states, or the...
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Research Article๐ท๐
Mesons in Asymmetric Hot and Dense Hadronic Matter
Divakar Pathak and Amruta Mishra
Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India
Correspondence should be addressed to Amruta Mishra; [email protected]
Received 8 June 2015; Revised 9 August 2015; Accepted 18 August 2015
Academic Editor: Juan Joseฬ Sanz-Cillero
Copyright ยฉ 2015 D. Pathak and A. Mishra. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited. The publication of this article was funded by SCOAP3.
The in-medium properties of ๐ท๐mesons are investigated within the framework of an effective hadronic model, which is a
generalization of a chiral ๐๐(3)model, to ๐๐(4), in order to study the interactions of the charmed hadrons. In the present work, the๐ท
๐mesons are observed to experience net attractive interactions in a dense hadronic medium, hence reducing themasses of the๐ท+
๐
and ๐ทโ๐mesons from the vacuum values. While this conclusion holds in both nuclear and hyperonic media, the magnitude of the
mass drop is observed to intensify with the inclusion of strangeness in the medium. Additionally, in hyperonic medium, the massdegeneracy of the ๐ท
๐mesons is observed to be broken, due to opposite signs of the Weinberg-Tomozawa interaction term in the
Lagrangian density. Along with the magnitude of the mass drops, the mass splitting between๐ท+๐and๐ทโ
๐mesons is also observed to
grow with an increase in baryonic density and strangeness content of the medium. However, all medium effects analyzed are foundto be weakly dependent on isospin asymmetry and temperature.We discuss the possible implications emanating from this analysis,which are all expected to make a significant difference to observables in heavy ion collision experiments, especially the upcomingCompressed Baryonic Matter (CBM) experiment at the future Facility for Antiproton and Ion Research (FAIR), GSI, where matterat high baryonic densities is planned to be produced.
1. Introduction
An effective description of hadronic matter is fairly commonin low-energyQCD [1โ3]. Realizing that baryons andmesonsconstitute the effective degrees of freedom in this regime,it is quite sensible to treat QCD at low-energies as aneffective theory of these quark bound states [1].This approachhas been vigorously pursued in various incarnations overthe years, with the different adopted strategies representingmerely different manifestations of the same underlying phi-losophy. The actual manifestations range from the quark-meson coupling model [4, 5], phenomenological, relativisticmean-field theories based on the Walecka model [6], alongwith their subsequent extensions, the method of QCD sumrules [7โ9], as well as the coupled channel approach [10, 11]for treating dynamically generated resonances, which hasfurther evolved into more specialized forms, namely, thelocal hidden gauge theory [12, 13], and formalisms based onincorporating heavy-quark spin symmetry (HQSS) [14, 15]into the coupled channel framework [16โ20]. Additionally,
the method of chiral-invariant Lagrangians [1] (which willalso be embraced in this work) has developed over theyears into a very successful strategy. This method uses aneffective field theoretical model in which the specific formof hadronic interactions is dictated by symmetry principlesand the physics governed predominantly by the dynamicsof chiral symmetryโits spontaneous breakdown implyinga nonvanishing scalar condensate โจ๐๐โฉ in vacuum. Onenaturally expects then that the hadrons composed of thesequarks would also be modified in accordance with thesecondensates [2, 3]. But while all hadrons would be subjecttomediummodifications from this perspective, pseudoscalarmesons have a special role in this context. In accordancewith Goldstoneโs theorem [1], spontaneous breaking of chiralsymmetry leads to the occurrence of massless pseudoscalarmodes, the so-called Goldstone bosons, which are generallyidentified with the spectrum of light pseudoscalar mesons,like the pions or kaons and antikaons [2, 3]. In a strict sense,however, none of these physical mesons is a true Goldstonemode, since they are all massive, while Goldstone modes
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015, Article ID 697514, 16 pageshttp://dx.doi.org/10.1155/2015/697514
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2 Advances in High Energy Physics
are supposed to be massless [21]. The origin of the massesof these mesons is related to the nonzero masses of lightquarks, as can be easily discerned from the Gell-Mann-Oakes-Renner (GOR) relations and, hence, from explicitsymmetry breaking terms (or explicit mass terms) in thechiral effective framework [1]. In fact, if one considers thelimiting situation of vanishing quark masses (๐๐ โ 0), themasses of these pseudoscalar mesons would also vanish,so that the perfect Goldstone modes are indeed recovered.For this reason, the physically observed light pseudoscalarmesons are dubbed pseudo-Goldstone bosons [21]. In purelythis sense, therefore, there is an inherent similarity betweenall these classes of pseudoscalar mesons; masses are acquiredthrough explicit quarkmass terms, and only themagnitude ofthese mass terms differs between these cases, being small forthe pions (since ๐
๐ข, ๐
๐< 10MeV), comparatively larger for
the kaons and antikaons (since ๐๐ โผ 150MeV), and appre-
ciably larger for the charmed pseudoscalar mesons. Thus,as we advance from the pions to the strange pseudoscalarmesons, with increasing mass, these mesons depart morefrom the ideal Goldstone mode character pertaining to thetheorem and there is considerable departure of the charmedpseudoscalar mesons from Goldstone mode behaviour dueto the explicit chiral symmetry breaking arising from thelarge charm quark mass (๐
๐โ 1.3GeV). An understanding
of the in-medium properties of the pseudoscalar mesonshas been an important topic of research, both theoreticallyand experimentally. Within the chiral effective approach,the pseudoscalar mesons are modified in the mediumdue to the modifications of the quark condensates in thehadronic medium. For pions, it is observed, however, thatmedium effects for them are weakened by the smallnessof explicit symmetry breaking terms [2, 3]. Considerablydetailed analysis of medium effects has been performedover the years, particularly in a chiral ๐๐(3) approach, forstrange pseudoscalar mesons (kaons and antikaons) [22โ25].For studying the charmed mesons, one needs to generalizethe ๐๐(3) model to ๐๐(4), in order to incorporate theinteractions of the charmedmesons to the light hadrons. Sucha generalization from ๐๐(3) to ๐๐(4) was initially done in[26], where the interaction Lagrangian was constructed forthe pseudoscalar mesons for ๐๐(4) from a generalization ofthe lowest order chiral ๐๐(3) Lagrangian. Since the chiralsymmetry is explicitly broken for the ๐๐(4) case due to thelargemass of the charmquark (๐
๐โ 1.3GeV), which ismuch
larger than the masses of the light quarks, for the study ofthe charmed (๐ท) pseudoscalar mesons [27โ29], we adopt thephilosophy of generalizing the chiral ๐๐(3)model to ๐๐(4) toderive the interactions of thesemesons with the light hadronsbut use the observed masses of these heavy hadrons as wellas empirical/observed values of their decay constants [30].With all these studies proving to be informative, the mostnatural direction of extension of this approach would be toanalyze these medium effects for a strange-charmed system(the ๐ท๐ mesons). Apart from pure theoretical interest, anunderstanding of the medium modifications of ๐ท๐ mesonsis important, since these can make a considerable differenceto experimental observables in the (ongoing and future)relativistic heavy ion collision experiments, besides being
significant in questions concerning their production andtransport in such experimental situations. For instance, in arecent work, He et al. [31] have shown that the modificationsof the ๐ท
๐meson spectrum can serve as a useful probe for
understanding key issues regarding hadronization in heavyion collisions. It is suggested that, by comparing observablesfor๐ท and๐ท๐ mesons, it is possible to constrain the hadronictransport coefficient.This comparison is useful since it allowsfor a clear distinction between hadronic and quark-gluonplasma behavior.
However, as far as the existing literature on this strange-charmed system of mesons is concerned, we observe thatonly the excited states of ๐ท๐ mesons have received con-siderable attention, predominantly as dynamically generatedresonances in various coupled channel frameworks [32,33]. One must bear in mind that, in certain situations, amolecular interpretation of these excited states (resonances)is more appropriate [34] for an explanation of their observed,larger than expected lifetimes. From this perspective, awhole plethora of possibilities have been entertained for theexcited๐ท
๐states, the standard quark-antiquark picture aside.
These include their description as molecular states, borneout of two mesons, four-quark states, or the still furtherexotic possibilitiesโas two-diquark states and as a mixtureof quark-antiquark and tetraquark states [32]. Prominentamong these is a treatment of๐ทโ
๐0(2317) as a๐ท๐พ bound state
[35, 36], ๐ท๐1(2460) as a dynamically generated ๐ทโ๐พ reso-
nance [37], and ๐ทโ๐2(2573) being treated within the hidden
local gauge formalism in coupled channel unitary approach[38, 39], as well as the vector ๐ทโ
๐states and the ๐ท+
๐(2632)
resonance treated in a multichannel approach [40]. Theformer three have also been covered consistently under thefour-quark picture [41]. So, while considerable attention hasbeen paid towards dynamically generating the higher excitedstates of the ๐ท๐ mesons, there is a conspicuous dearth [42]of available information about the medium modifications ofthe lightest pseudoscalar ๐ท๐ mesons, ๐ท๐(1968.5), the onethat we know surely is well described within the quark-antiquark picture. In fact, to the best of our knowledge, theentire existing literature about the (๐ฝ
๐= 0
โ)๐ท
๐mesons
in a hadronic medium is limited to the assessment of theirspectral distributions andmedium effects on the dynamicallygenerated resonances borne out of the interaction of these๐ท
๐mesons with other hadron species, in the coupled channel
analyses of [42โ45]. This situation is quite unlike their open-charm, non-strange counterparts, the๐ทmesons, which boastof a sizeable amount of literature having been extensivelyinvestigated using a multitude of approaches over the years.In stark contrast, the available literature concerning the in-medium behavior of pseudoscalar ๐ท
๐mesons can at best be
described as scanty, and there is need for more work on thissubject. If one considers this problem from the point of viewof the (extended) chiral effective approach, this scantinessis most of all because of the lack of a proper frameworkwhere the relevant form of the interactions for the๐ท๐ mesonswith the light hadrons (or in more generic terms, of meson-baryon interactions with the charm sector covered), basedon arguments of symmetry and invariance, could be writtendown. Clearly, such interactions would have to be based on
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Advances in High Energy Physics 3
๐๐(4) symmetry and bear all these pseudoscalar mesons andbaryons in 15-plet and 20-plet representations, respectively,withmeson-baryon interaction terms still in accordance withthe general framework for writing chiral-invariant structures,as well as bearing appropriate symmetry breaking termsobeying the requisite transformation behavior under chiraltransformations [1], which is quite a nontrivial problem. Oflate, such formalisms have been proposed in [43, 46], as anextension of the frameworks based on chiral Lagrangians,where ๐๐(4) symmetry forms the basis for writing downthe relevant interaction terms. However, since the mass ofthe charm quark is approximately 1.3 GeV [47], which isconsiderably larger than that of the up, down, and strangequarks, the ๐๐(4) symmetry is explicitly broken by this largecharm quark mass. Hence, this formalism only uses thesymmetry to derive the form of the interactions, whereasan explicit symmetry breaking term accounts for the largequark mass through the introduction of mass terms of therelevant (๐ท or ๐ท
๐) mesons. Also, ๐๐(4) symmetry being
badly broken implies that any symmetry and order in themasses and decay constants, as predicted on the basis of๐๐(4) symmetry, would not hold in reality. The same isacknowledged in this approach [46] and, as has been alreadymentioned, one does not use the masses and decay constantsas expected on the basis of ๐๐(4) symmetry but rather theirobserved Particle Data Group (PDG) [47] values. Overall,therefore, ๐๐(4) symmetry is treated (appropriately) as beingbroken in this approach. Also, it is quite well-known, throughboth model-independent [48] and model-dependent [29]calculations, that the light quark condensates (โจ๐ข๐ขโฉ, โจ๐๐โฉ) aremodified significantly in a hadronic medium with mediumparameters like density and temperature; the strange quarkcondensate โจ๐ ๐ โฉ is comparatively stolid and its variation issignificantly more subdued, while upon advancing to thecharm sector the variation in the charmed quark condensateโจ๐๐โฉ is altogether negligible in the entire hadronic phase [48].These observations form the basis for treating the charmdegrees of freedom of open-charm pseudoscalar mesons asfrozen in the medium, as was the case in the treatments of[28, 29, 46].Thus, as we advance from pions and kaons to thecharmed pseudoscalar mesons, the generalization is perfectlynatural but with the aforementioned caveats. Provided allthese aspects are taken into account, a generalization ofthis chiral effective framework to open-charm pseudoscalarmesons is quite reasonable and sane, and the predictions fromsuch an extended chiral effective approach bode very well[28] with alternative calculations based on the QCD sum ruleapproach, quark-meson coupling model, coupled channelapproach, and studies of quarkonium dissociation usingheavy-quark potentials from lattice QCD at finite tempera-tures. Additionally, it is interesting to note that this approach,followed in [28, 29, 46] for the charmedpseudoscalarmesons,has recently been extended to the bottom sector and used tostudy the medium behavior of the open bottom pseudoscalar๐ต, ๐ต, and ๐ต
๐mesons [49, 50]. The inherent philosophy
beneath this extension continues to be the same; the dynam-ics of the heavy quark/antiquark is treated as frozen, and theinteractions of the light quark (or antiquark) of the meson,with the particles constituting the medium, are responsible
for the medium modifications. With this subsequent gener-alization as well, the physics of the medium behavior thatfollows from this approach is in agreement [49] with worksbased on alternative, independent approaches, like the heavymeson effective theory, quark-meson coupling model, andthe QCD sum rule approach. Thus, these aforementioned,prior works based on the generalization of the original chiraleffective approach to include heavy flavored mesons, aretotally concordant with results from alternative approachesfollowed in the literature, which lends an aura of credibilityto this strategy. Given this backdrop, it is clear that theseformalisms wipe out the reason why such an investigation forthe๐ท๐ mesons within the effective hadronic model, obtainedby generalizing the chiral ๐๐(3)model to ๐๐(4), has not beenundertaken till date and permit this attempt to fill the void.
We organize this paper as follows. In Section 2, we outlinethe chiral ๐๐(3)
๐ฟร ๐๐(3)
๐ model (and its generalization
to the ๐๐(4) case) used in this investigation. In Section 3,the Lagrangian density for the ๐ท
๐mesons, within this
extended framework, is explicitly written down and is usedto derive their in-medium dispersion relations. In Section 4,we describe and discuss our results for the in-mediumproperties of ๐ท
๐mesons, first in the nuclear matter case and
then in the hyperonic matter situation, following which webriefly discuss the possible implications of these mediummodifications. Finally, we summarize the entire investigationin Section 5.
2. The Effective Hadronic Model
As mentioned previously, this study is based on a general-ization of the chiral ๐๐(3)๐ฟ ร ๐๐(3)๐ model [51], to ๐๐(4).We summarize briefly the rudiments of the model, whilereferring the reader to [51, 52] for the details. This is aneffective hadronic model of interacting baryons and mesons,based on a nonlinear realization of chiral symmetry [53, 54],where chiral invariance is used as a guiding principle, indeciding the form of the interactions [55โ57]. Additionally,the model incorporates a scalar dilaton field, ๐, to mimic thebroken scale invariance of QCD [52]. Once these invariancearguments determine the form of the interaction terms, oneresorts to a phenomenological fitting of the free parametersof the model, to arrive at the desired effective Lagrangiandensity for these hadron-hadron interactions. The generalexpression for the chiral model Lagrangian density reads
L =Lkin +โ๐
LBW +Lvec +L0 +Lscale break
+LSB.
(1)
In (1),Lkin is the kinetic energy term, whileLBW denotes thebaryon-meson interaction term. Here, baryon-pseudoscalarmeson interactions generate the baryon masses. Lvec treatsthe dynamical mass generation of the vector mesons throughcouplings with scalar mesons. The self-interaction terms ofthese mesons are also included in this term.L
0contains the
meson-meson interaction terms, which induce spontaneousbreaking of chiral symmetry. Lscale break introduces scaleinvariance breaking, via a logarithmic potential term in the
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4 Advances in High Energy Physics
scalar dilaton field, ๐. Finally, LSB refers to the explicitsymmetry breaking term. This approach has been employedextensively to study the in-medium properties of hadrons,particularly pseudoscalar mesons [22โ25]. As was observedin Section 1 as well, this would be most naturally extended tothe charmed (nonstrange and strange) pseudoscalar mesons.However, that calls for this chiral ๐๐(3) formalism to begeneralized to ๐๐(4), which has been addressed in [43,46]. For studying the in-medium behavior of pseudoscalarmesons, the following contributions need to be analyzed[28, 29, 46]:
L =LWT +L1st Range +L๐1
+L๐2
+LSME. (2)
In (2),LWT denotes theWeinberg-Tomozawa term, given bythe expression [46]
LWT = โ1
2[๐ต
๐๐๐๐พ
๐((ฮ
๐)
๐
๐
๐ต๐๐๐+ 2(ฮ
๐)
๐
๐
๐ต๐๐๐)] , (3)
with repeated indices summed over. Baryons are representedby the tensor ๐ต๐๐๐, which is antisymmetric in its first twoindices [43]. The indices ๐, ๐, and ๐ run from 1 to 4, and onecan directly read the quark content of a baryon state, withthe following identification: 1 โ ๐ข, 2 โ ๐, 3 โ ๐ , 4 โ ๐.However, the heavier, charmed baryons are discounted fromthis analysis. In (3), ฮ
๐is defined as
ฮ๐
= โ๐
4(๐ข
โ (๐๐๐ข) โ (๐๐๐ข
โ ) ๐ข + ๐ข (๐๐๐ข
โ ) โ (๐๐๐ข) ๐ข
โ ) ,
(4)
where the unitary transformation operator, ๐ข = exp(๐๐๐พ5/โ2๐0), is defined in terms of the matrix of pseudoscalarmesons, ๐ = (๐๐๐๐/โ2), ๐๐ representing the generalizedGell-Mann matrices. Further, LSME is the scalar mesonexchange term, which is obtained from the explicit symmetrybreaking term
LSB = โ1
2Tr (๐ด
๐(๐ข๐๐ข + ๐ข
โ ๐๐ข
โ )) , (5)
where ๐ด๐ = (1/
โ2) diag[๐2๐๐๐, ๐
2
๐๐๐, (2๐
2
๐พ๐๐พ โ ๐
2
๐๐๐),
(2๐2
๐ท๐
๐ทโ๐
2
๐๐
๐)] and๐ refers to the scalar meson multiplet
[28]. Also, the first range term is obtained from the kineticenergy term of the pseudoscalar mesons and is given by theexpression
L1st Range = Tr (๐ข๐๐๐ข
๐๐ + ๐๐ข
๐๐ข
๐๐) , (6)
where ๐ข๐= โ๐((๐ข
โ (๐
๐๐ข) โ (๐
๐๐ข
โ )๐ข) โ (๐ข(๐
๐๐ข
โ ) โ (๐
๐๐ข)๐ข
โ ))/4.
Lastly, the ๐1and ๐
2range terms are
L๐1
=๐
1
4(๐ต๐๐๐๐ต
๐๐๐(๐ข๐)
๐
๐
(๐ข๐)
๐
๐) , (7)
L๐2
=๐
2
2[๐ต๐๐๐(๐ข๐)
๐
๐
((๐ข๐)
๐
๐๐ต
๐๐๐+ 2(๐ข
๐)
๐
๐๐ต
๐๐๐)] . (8)
Adopting the mean-field approximation [6, 52], the effectiveLagrangian density for scalar and vector mesons simplifies;
the same is used subsequently to derive the equations ofmotion for the nonstrange scalar-isoscalar meson ๐, scalar-isovector meson ๐ฟ, and strange scalar meson ๐ and for thevector-isovector meson ๐, nonstrange vector meson ๐, andstrange vector meson ๐, within this model.
The ๐ท๐meson interaction Lagrangian density and in-
medium dispersion relations, as they follow from the abovegeneral formulation, are described next.
3. ๐ท๐
Mesons in Hadronic Matter
The Lagrangian density for the ๐ท๐ mesons in isospin-asym-metric, strange, hadronic medium is given as
Ltotal =Lfree +Lint. (9)
ThisLfree is the free Lagrangian density for a complex scalarfield (which corresponds to the ๐ท
๐mesons in this case) and
reads
Lfree = (๐๐๐ท
+
๐) (๐๐๐ท
โ
๐) โ ๐
2
๐ท๐
(๐ท+
๐๐ท
โ
๐) . (10)
On the other hand,Lint is determined to be
Lint = โ๐
4๐2
๐ท๐
[(2 (ฮ0
๐พ๐ฮ
0+ ฮ
โ
๐พ๐ฮ
โ) + ฮ
0
๐พ๐ฮ
0
+ ฮฃ+
๐พ๐ฮฃ
++ ฮฃ
0
๐พ๐ฮฃ
0+ ฮฃ
โ
๐พ๐ฮฃ
โ) (๐ท
+
๐(๐
๐๐ท
โ
๐)
โ (๐๐๐ท
+
๐)๐ท
โ
๐)] +
๐2
๐ท๐
โ2๐๐ท๐
[(๐+ ๐
๐) (๐ท
+
๐๐ท
โ
๐)]
โโ2
๐๐ท๐
[(๐+ ๐
๐) ((๐
๐๐ท
+
๐) (๐
๐๐ท
โ
๐))] +
๐1
2๐2
๐ท๐
[(๐๐
+ ๐๐ + ฮ0
ฮ0+ ฮฃ
+
ฮฃ++ ฮฃ
0
ฮฃ0+ ฮฃ
โ
ฮฃโ+ ฮ
0
ฮ0
+ ฮโ
ฮโ) ((๐
๐๐ท
+
๐) (๐
๐๐ท
โ
๐))]
+๐
2
2๐2
๐ท๐
[(2 (ฮ0
ฮ0+ ฮ
โ
ฮโ) + ฮ
0
ฮ0+ ฮฃ
+
ฮฃ+
+ ฮฃ0
ฮฃ0+ ฮฃ
โ
ฮฃโ) ((๐
๐๐ท
+
๐) (๐
๐๐ท
โ
๐))] .
(11)
In this expression, the first term (with coefficient โ๐/4๐2๐ท๐
)is the Weinberg-Tomozawa term, obtained from (3), thesecond term (with coefficient ๐2
๐ท๐
/โ2๐๐ท๐
) is the scalarmeson exchange term, obtained from the explicit symmetrybreaking term of the Lagrangian (see (5)), the third term(with coefficient โโ2/๐
๐ท๐
) is the first range term (see (6)),and the fourth and fifth terms (with coefficients (๐
1/๐
2
๐ท๐
) and(๐
2/๐2
๐ท๐
), resp.) are the ๐1 and ๐2 terms, calculated from (7)and (8), respectively. Also, ๐ = ๐ โ ๐
0is the fluctuation of
the strange scalar field from its vacuum value.Themean-fieldapproximation, mentioned earlier, is a useful, simplifying
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Advances in High Energy Physics 5
measure in this context, since it permits us to have the fol-lowing replacements:
๐ต๐๐ต
๐โ โจ๐ต
๐๐ต
๐โฉ โก ๐ฟ
๐๐๐
๐
๐,
๐ต๐๐พ
๐๐ต
๐โ โจ๐ต
๐๐พ
๐๐ต
๐โฉ = ๐ฟ
๐๐(๐ฟ
0
๐(๐ต
๐๐พ
๐๐ต
๐)) โก ๐ฟ
๐๐๐
๐.
(12)
Thus, the interaction Lagrangian density can be recast interms of the baryonic number densities and scalar densities,given by the following expressions:
๐๐=๐พ
๐
(2๐)3โซ๐
3๐(
1
exp ((๐ธโ๐(๐) โ ๐
โ
๐) /๐) + 1
โ1
exp ((๐ธโ๐(๐) + ๐
โ
๐) /๐) + 1
) ,
(13)
๐s๐=๐พ
๐
(2๐)3โซ๐
3๐๐
โ
๐
๐ธโ
๐(๐)(
1
exp ((๐ธโ๐(๐) โ ๐
โ
๐) /๐) + 1
+1
exp ((๐ธโ๐(๐) + ๐
โ
๐) /๐) + 1
) .
(14)
In the above, ๐โ๐and ๐โ
๐are the effective mass and effective
chemical potential of the ๐th baryon, given as๐โ๐= โ(๐
๐๐๐ +
๐๐๐๐ + ๐
๐ฟ๐๐ฟ), ๐โ
๐= ๐
๐โ (๐
๐๐๐
3๐ + ๐
๐๐๐ + ๐
๐๐๐), ๐ธโ
๐(๐) =
(๐2+ ๐
โ
๐
2)
1/2, and ๐พ๐ = 2 is the spin degeneracy factor. One
can find the equations of motion for the ๐ท+๐and ๐ทโ
๐mesons,
by the use of Euler-Lagrange equations on this Lagrangiandensity. The linearity of these equations follows from (11),which allows us to assume plane wave solutions (โผ๐๐(๏ฟฝโ๏ฟฝโ โ๐โ๐๐ก)),and hence, โFourier transformโ these equations, to arrive atthe in-medium dispersion relations for the๐ท๐ mesons.Thesehave the general form
โ๐2+ ๏ฟฝโ๏ฟฝ
2+ ๐
2
๐ท๐
โ ฮ (๐,๏ฟฝโ๏ฟฝ) = 0, (15)
where๐๐ท๐
is the vacuummass of the๐ท๐mesons andฮ (๐, |๏ฟฝโ๏ฟฝ|)
is the self-energy of the๐ท๐mesons in the medium. Explicitly,
the latter reads
ฮ (๐,๏ฟฝโ๏ฟฝ) = [(
๐1
2๐2
๐ท๐
(๐๐
๐+ ๐
๐
๐+ ๐
๐
ฮ+ ๐
๐
ฮฃ+ + ๐
๐
ฮฃ0 + ๐
๐
ฮฃโ
+ ๐๐
ฮ0 + ๐
๐
ฮโ)) + (
๐2
2๐2
๐ท๐
(2 (๐๐
ฮ0 + ๐
๐
ฮโ) + ๐
๐
ฮ+ ๐
๐
ฮฃ+
+ ๐๐
ฮฃ0 + ๐
๐
ฮฃโ)) โ (
โ2
๐๐ท๐
(๐+ ๐
๐))] (๐
2โ ๏ฟฝโ๏ฟฝ
2)
ยฑ [1
2๐2
๐ท๐
(2 (๐ฮ0 + ๐
ฮโ) + ๐
ฮ+ ๐
ฮฃ+ + ๐
ฮฃ0 + ๐
ฮฃโ)]๐
+ [
๐2
๐ท๐
โ2๐๐ท๐
(๐+ ๐
๐)] ,
(16)
where the + and โ signs in the coefficient of ๐ refer to ๐ท+๐
and ๐ทโ๐, respectively, and we have used (12) to simplify the
bilinears. In the rest frame of these mesons (i.e., setting |๏ฟฝโ๏ฟฝ| =0), these dispersion relations reduce to
โ๐2+ ๐
2
๐ท๐
โ ฮ (๐, 0) = 0, (17)
which is a quadratic equation in ๐, that is, of the form ๐ด๐2 +๐ต๐ + ๐ถ = 0, where the coefficients ๐ด, ๐ต, and ๐ถ depend onvarious interaction terms in (15) and read
๐ด = [1 + (๐
1
2๐2
๐ท๐
โ
(๐+๐ป)
๐๐
๐)
+ (๐
2
2๐2
๐ท๐
(2โ
ฮ
๐๐
๐+ โ
(๐ปโฮ)
๐๐
๐))
โ (โ2
๐๐ท๐
(๐+ ๐
๐))] ,
(18)
๐ต = ยฑ[1
2๐2
๐ท๐
(2โ
ฮ
๐๐ + โ
(๐ปโฮ)
๐๐)] , (19)
๐ถ = [โ๐2
๐ท๐
+
๐2
๐ท๐
โ2๐๐ท๐
(๐+ ๐
๐)] . (20)
As before, the โ+โ and โโโ signs in ๐ต correspond to ๐ท+๐and
๐ทโ
๐, respectively. In writing the above summations, we have
used the following notation:๐ปdenotes the set of all hyperons,๐ is the set of nucleons, ฮ represents the Xi hyperons (ฮโ,0),and (๐ปโฮ) denotes all hyperons other than Xi hyperons, thatis, the set of baryons (ฮ, ฮฃ+,โ,0) which all carry one strangequark. This form is particularly convenient for later analysis.Also, the optical potential of๐ท๐ mesons is defined as
๐ (๐) = ๐ (๐) โ โ๐2 + ๐2
๐ท๐
, (21)
where ๐ (=|๏ฟฝโ๏ฟฝ|) refers to the momentum of the respective ๐ท๐
meson, and ๐(๐) represents its momentum-dependent in-medium energy.
In the next section, we study the sensitivity of the ๐ท๐
meson effective mass on various characteristic parameters ofhadronic matter, namely, baryonic density (๐
๐ต), temperature
(๐), isospin asymmetry parameter ๐ = โโ๐๐ผ
3๐๐
๐/๐
๐ต, and the
strangeness fraction๐๐ = โ
๐|๐
๐|๐
๐/๐
๐ต, where ๐
๐and ๐ผ
3๐denote
the strangeness quantum number and the third componentof isospin of the ๐th baryon, respectively.
4. Results and Discussion
Before describing the results of our analysis of ๐ท๐ mesonsin a hadronic medium, we first discuss our parameter choiceand the various simplifying approximations employed in thisinvestigation.The parameters of the effective hadronic modelare fitted to the vacuummasses of baryons, nuclear saturationproperties, and other vacuum characteristics in the mean-field approximation [51, 52]. In this investigation, we haveused the same parameter set that has earlier been used to
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6 Advances in High Energy Physics
study charmed (๐ท) mesons within this effective hadronicmodel [46]. In particular, we use the same values of theparameters ๐
1and ๐
2of the range terms (๐
1= 2.56/๐
๐พand
๐2= 0.73/๐
๐พ), fitted to empirical values of kaon-nucleon
scattering lengths for the ๐ผ = 0 and ๐ผ = 1 channels, asemployed in earlier treatments [22, 23, 28, 46]. For an exten-sion to the strange-charmed system, the only extra parameterthat needs to be fitted is the ๐ท๐ meson decay constant, ๐๐ท
๐
,which is treated as follows. By extrapolating the results of [57],we arrive at the following expression for ๐๐ท
๐
in terms of thevacuum values of the strange and charmed scalar fields:
๐๐ท๐
=โ (๐
0+ ๐
๐0)
โ2. (22)
For fitting the value of ๐๐ท๐
, we retain the same values of ๐0and ๐
0as in the earlier treatments of this chiral effectivemodel
[23, 51] and determine ๐๐0from the expression for๐
๐ทin terms
of ๐0and ๐
๐0[28], using the Particle Data Group (PDG) value
of๐๐ท= 206MeV. Substituting these in (22), our fitted value of
๐๐ท๐
comes out to be 235MeV, which is close to its PDG valueof 260MeV and, particularly, is of the same order as typicallattice QCD calculations for the same [47]. We therefore per-sist with this fitted value ๐
๐ท๐
= 235MeV in this investigation.Also, we treat the charmed scalar field (๐
๐) as being arrested at
its vacuum value (๐๐0) in this investigation, as has been con-
sidered in other treatments of charmedmesons in a hadroniccontext [28, 46]. This neglect of charm dynamics appearsnatural from a physical viewpoint, owing to the large massof a charm quark (๐๐ โผ 1.3GeV) [47]. The same was verifiedin an explicit calculation in [48], where the charm condensatewas observed to vary very weakly in the temperature range ofinterest to us in this regime [58โ60]. As our last approxima-tion, we point out that, in the current investigation, we workwithin the โfrozen glueball limitโ [52], where the scalar dilatonfield (๐) is regarded as being frozen at its vacuum value (๐
0).
This approximation was relaxed in a preceding work [29]within this effective hadronic model, where the in-mediummodifications of this dilaton field were found to be quitemeager. We conclude, therefore, that this weak dependenceonly serves to justify the validity of this assumption.
Wenext describe our analysis for the in-mediumbehaviorof๐ท
๐mesons, beginning first with the nuclearmatter (๐
๐ = 0)
situation and including the hyperonic degrees of freedomonly later.This approach has the advantage thatmany featuresof the ๐ท
๐meson in-medium behavior, common between
these regimes, are discussed in detail in a more simplifiedcontext, and the effect of strangeness becomes a lot clearer.
In nuclear matter, the Weinberg-Tomozawa term and the๐
2range term vanish, since they depend on the number
densities and scalar densities of the hyperons, and have nocontribution from the nucleons. It follows from the self-energy expression that only the Weinberg-Tomozawa termdiffers between ๐ท+
๐and ๐ทโ
๐; all other interaction terms are
absolutely identical for them. Thus, a direct consequence ofthe vanishing of Weinberg-Tomozawa contribution is that๐ท
+
๐and ๐ทโ
๐are degenerate in nuclear matter. As mentioned
earlier, the๐ท๐meson dispersion relations, with |๏ฟฝโ๏ฟฝ| = 0, takes
the quadratic equation form (๐ด๐2 + ๐ต๐ + ๐ถ = 0), where
the coefficients (given by (18)โ(20)) reduce to the followingexpressions in nuclear matter:
๐ด(๐)
= [1 + (๐
1
2๐2
๐ท๐
(๐๐
๐+ ๐
๐
๐)) โ (
โ2
๐๐ท๐
(๐+ ๐
๐))] ,
(23)
๐ต(๐)= 0, (24)
๐ถ(๐)= [โ๐
2
๐ท๐
+
๐2
๐ท๐
โ2๐๐ท๐
(๐+ ๐
๐)] , (25)
where the subscript (๐) emphasizes the nuclear mattercontext. For solving this quadratic equation, we require thevalues of ๐, as well as the scalar densities of protons andneutrons, which are obtained from a simultaneous solutionof coupled equations of motion for the scalar fields, subjectto constraints of fixed values of ๐
๐ตand ๐. The behavior
of the scalar fields, thus obtained, has been discussed indetail in [29]. Here, we build upon these scalar fields andproceed to discuss the behavior of solutions of the in-mediumdispersion relations of ๐ท
๐mesons, given by (23) to (25). The
variation of the ๐ท๐ meson in-medium mass, ๐(๏ฟฝโ๏ฟฝ = 0), with
baryonic density in nuclear matter, along with the individualcontributions to this net variation, is shown in Figure 1 forboth zero and finite temperature value (๐ = 100MeV). It isobserved that the in-medium mass of ๐ท
๐mesons decreases
with density, while being weakly dependent on temperatureand isospin asymmetry parameter. We can understand theobserved behavior through the following analysis. From (23)to (25), we arrive at the following closed-form solution for the๐ท
๐meson effective mass in nuclear matter:
๐
= ๐๐ท๐
[
[
1 โ (๐+ ๐
๐) /โ2๐
๐ท๐
1 + (๐1/2๐2
๐ท๐
) (๐๐ ๐+ ๐๐
๐) โ (โ2/๐๐ท
๐
) (๐ + ๐๐)
]
]
1/2
.
(26)
From this exact closed-form solution, several general con-clusions regarding the in-medium behavior of ๐ท
๐mesons in
nuclear matter can be drawn. On the basis of this expression,the ๐
1range term appearing in the denominator will lead to a
decrease in themediummass with increase in density.The in-medium mass is, additionally, also modified by the mediumdependence of ๐ (we assume the value of ๐๐ to be frozen atits vacuum value and hence ๐
๐= 0). However, the density
dependence of ๐ is observed to be quite subdued in nuclearmatter. The total change in the value of ๐ from its vacuumvalue is observed; that is, (๐)max. โ 15MeV. This value isacquired at a baryonic density of ๐๐ต โ 3๐0 and thereafter itappears to saturate. This behavior of the strange scalar field,though ubiquitous in this chiral model [29, 51], is not justlimited to it. Even in calculations employing the relativisticHartree approximation to determine the variation of thesescalar fields [61, 62], a behavior similar to this mean-fieldtreatment is observed. We also point out that the saturationof the scalar meson exchange and first range term follows
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Advances in High Energy Physics 7
1800
1850
1900
1950
2000
0 1 2 3 4 5 6
TotalScalar
Range
(๐B/๐0)
Ener
gy๐(k=0)
(MeV
)
(a) ๐ = 0
1800
1850
1900
1950
2000
0 1 2 3 4 5 6
TotalScalar
Range
(๐B/๐0)
Ener
gy๐(k=0)
(MeV
)
(b) ๐ = 100MeV
Figure 1:The various contributions to the๐ท๐meson energies, ๐(๏ฟฝโ๏ฟฝ = 0), in nuclear matter, plotted as functions of baryonic density in units of
the nuclear saturation density (๐๐ต/๐
0). In each case, the isospin asymmetric situation (๐ = 0.5), as described in the legend, is also compared
against the symmetric situation (๐ = 0), represented by dotted lines.
as a direct consequence of the saturation of ๐, as is impliedby their proportionality in (11) and (25). At larger densities,while the terms having ๐ saturate, the contribution from the๐
1range term continues to rise. With ๐ saturating at a value
of around 15MeV and the value of ๐๐ท๐
chosen to be 235MeVin the present investigation, the last term in the numeratoras well as the denominator in (26) turns out to be muchsmaller as compared to unity. Moreover, the ๐
1term in the
denominator in the expression for the in-mediummass of๐ท๐
given by (26) also turns out to be much smaller as comparedto unity, with (๐
1(๐
๐
๐+ ๐
๐
๐)/2๐
2
๐ท๐
) โ 0.05 at a density of๐
๐ต= 6๐
0. Hence the smallness of this term as compared
to unity is justified for the entire range of densities we areconcerned with, in the present investigation. In order to readmore into our solution, we expand the argument of the squareroot, in a binomial series (assuming (๐ +๐
๐)/โ2๐๐ท
๐
โช 1 and(๐
1(๐
๐
๐+๐
๐
๐)/2๐
2
๐ท๐
) โช 1), and retain up to only the first-orderterms. This gives, as the approximate solution,
๐ โ ๐๐ท๐
[1 +
(๐+ ๐
๐)
2โ2๐๐ท๐
โ๐
1
4๐2
๐ท๐
(๐๐
๐+ ๐
๐
๐)] . (27)
Moreover, since ๐๐ ๐โ ๐๐ at small densities, the ๐1 range term
contribution in the above equation approximately equals(๐
1/4๐
2
๐ท๐
)๐๐ต
at low densities. The first of the terms inthe approximate expression given by (27) is an increasingfunction while the second one is a decreasing function ofdensity. Their interplay generates the observed curve shape,the repulsive contribution being responsible for the smallhump in an otherwise linear fall, at small densities (0 < ๐
๐ต<
0.5๐0). While the attractive ๐1 term would have produced alinear decrease right away, the role of repulsive contributionis to impede this decrease, hence producing the hump. Atmoderately higher densities, however, the contribution fromthe second term outweighs the first, which is why we see alinear drop with density. This is observed to be the case, tillaround ๐
๐ตโ 2๐
0. At still larger densities, the approximation
(๐๐ ๐โ ๐
๐) breaks down, though our binomial expansion
is still valid. Since scalar density falls slower than numberdensity, the term [โ(๐
1/4๐
2
๐ท๐
)(๐๐
๐+ ๐
๐
๐)] will fall faster than
[โ(๐1/4๐
2
๐ท๐
)๐๐ต]. This is responsible for the change in slope
of the curve at intermediate densities, where a linear fallwith density is no longer obeyed. So, at intermediate andlarge densities, the manner of the variation is dictated bythe scalar densities, in the ๐1 range term. From (18) to (20),one would expect the mass modifications of ๐ท๐ mesons tobe insensitive to isospin asymmetry, since, for example, inthis nuclear matter case, the dispersion relation bears isospinsymmetric terms like (๐๐
๐+ ๐
๐
๐). (This is in stark contrast with
earlier treatments of kaons and antikaons [22, 23] as wellas the ๐ท mesons [28] within this effective model, where thedispersion relations had terms like (๐๐
๐โ๐
๐
๐) or (๐
๐โ๐
๐), which
contributed to asymmetry.) However, the๐ท๐meson effective
mass is observed to depend on asymmetry in Figure 1, thoughthe dependence is weak. For example, the values of ๐(๐ =0), for the isospin symmetric situation (๐ = 0), are 1913,1865, and 1834MeV, respectively, at ๐
๐ต= 2๐
0, 4๐
0, and
6๐0, while the corresponding numbers for the (completely)
asymmetric (๐ = 0.5) situation are 1917, 1875, and 1849MeV,respectively, at ๐ = 0. Since the ๐-dependence of LSME and
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8 Advances in High Energy Physics
L1st Range contributions must be identical, one can reason
from Figure 1 that this isospin dependence of ๐ท๐meson
effective mass is almost entirely due to the ๐1range term
(โผ(๐๐ ๐+ ๐
๐
๐)), which was expected to be isospin symmetric.
This apparently counterintuitive behavior has been observedearlier in [29], in the context of๐ทmesons.This is because thevalue of (๐๐
๐+ ๐
๐
๐) turns out to be different for symmetric and
asymmetric situations, contrary to naive expectations. Sincethe scalar-isovector ๐ฟ meson (๐ฟ โผ โจ๐ข๐ข โ ๐๐โฉ) is responsiblefor introducing isospin asymmetry in this effective hadronicmodel [51], owing to the equations of motion of the scalarfields being coupled, the values of the other scalar fields turnout to be different in the symmetric (๐ฟ = 0) and asymmetric(๐ฟ ฬธ= 0) cases. The same is also reflected in the values ofthe scalar densities calculated from these scalar fields, whichleads to the observed behavior.
Additionally, we observe from a comparison of Figures1(a) and 1(b) that the magnitude of the ๐ท
๐meson mass drop
decreases with an increase in temperature from ๐ = 0 to๐ = 100MeV. For example, in the symmetric (๐ = 0)situation, at ๐ = 0, the ๐ท
๐meson mass values, at ๐
๐ต=
2๐0, 4๐
0, and 6๐
0, are 1913, 1865, and 1834MeV, respectively,
which grow to 1925, 1879, and 1848MeV, respectively, at ๐ =100MeV. Likewise, with ๐ = 0.5, the corresponding valuesread 1917, 1875, and 1849MeV at ๐ = 0, while the samenumbers, at ๐ = 100MeV, are 1925, 1883, and 1855MeV.Thus, though small, there is a definite reduction in themagnitude of the mass drops, as we go from ๐ = 0 to๐ = 100MeV, in each of these cases. This behavior canbe understood, from the point of view of the temperaturevariation of scalar condensates, in the following manner. Itis observed that the scalar fields decrease with an increasein temperature from ๐ = 0 to 100MeV [27, 29]. In [27],the same effect was understood as an increase in the nucleonmass with temperature. The equity of the two argumentscan be seen by invoking the expression relating the baryonmass to the scalar fieldsโ magnitude, ๐โ
๐= โ(๐
๐๐๐ + ๐
๐๐๐ +
๐๐ฟ๐๐ฟ) [51]. The temperature dependence of the scalar fields
(๐, ๐, ๐ฟ) has been studied within the model in [29], whichare observed to be different for the zero and finite baryondensities. At zero baryon density, themagnitudes of the scalarfields ๐ and ๐ are observed to remain almost constant upto a temperature of about 125MeV, above which these areobserved to decrease with temperature. This behaviour canbe understood from the expression of ๐๐
๐ given by (14) for the
situation of zero density, that is, for ๐โ๐= 0, which decreases
with increase in temperature. The temperature dependenceof the scalar density in turn determines the behaviour of thescalar fields. The scalar fields which are solved from theirequations of motion behave in a similar manner as the scalardensity. At finite densities, that is, for nonzero values of theeffective chemical potential, ๐โ
๐, however, the temperature
dependence of the scalar density is quite different fromthe zero density situation. For finite baryon density, withincrease in temperature, there are contributions also fromhigher momenta, thereby increasing the denominator of theintegrand on the right-hand side of the baryon scalar densitygiven by (14). This leads to a decrease in the scalar density. Atfinite baryon densities, the competing effects of the thermal
distribution functions and the contributions from the highermomentum states give rise to the temperature dependenceof the scalar density, which in turn determine the behaviourof the ๐ and ๐ fields with temperature. These scalar fieldsare observed to have an initial increase in their magnitudesup to a temperature of around 125โ150MeV, followed by adecrease as the temperature is further raised [29].This kind ofbehavior of the scalar ๐ field on temperature at finite densitieshas also been observed in the Walecka model in [63]. In fact,we point out that a decrease in the scalar condensates with anincrease in temperature, though small in the hadronic regime(
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Advances in High Energy Physics 9
0 1 2 3 4 5 61800
1850
1900
1950
2000
TotalWT
ScalarRange
(๐B/๐0)
Ener
gy๐(k=0)
(MeV
)
(a) ๐ท+๐
0 1 2 3 4 5 61800
1850
1900
1950
2000
TotalWT
ScalarRange
(๐B/๐0)
Ener
gy๐(k=0)
(MeV
)
(b) ๐ทโ๐
Figure 2: The various contributions to the effective mass of ๐ท+๐and ๐ทโ
๐mesons, as a function of baryonic density, for typical values of
temperature (๐ = 100MeV) and strangeness fraction (๐๐ = 0.5). In each case, the asymmetric situation (๐ = 0.5), as described in the legend,
is also compared against the symmetric situation (๐ = 0), represented by dotted lines.
degeneracy of ๐ท+๐and ๐ทโ
๐now stands broken. For example,
the values of (๐๐ท+
๐
, ๐๐ทโ
๐
) at ๐๐ต= ๐
0, 2๐
0, 4๐
0, and 6๐
0
are observed to be (1948, 1953), (1918, 1928), (1859, 1879),and (1808, 1838)MeV, respectively, as can be seen from thefigure. Thus, except at vacuum (๐
๐ต= 0), mass difference of
๐ท+
๐and ๐ทโ
๐is nonzero at finite ๐
๐ต, growing in magnitude
with density. This mass degeneracy breaking is a directconsequence of nonzero contributions from the Weinberg-Tomozawa term, which follows from (11), (16), and (19). Thesame may be reconciled with Figure 2, from which it followsthat, except for this Weinberg-Tomozawa term acquiringequal and opposite values for these mesons, all other termsare absolutely identical for them. Once again, on the basis ofthe following analysis of the ๐ท
๐meson dispersion relations
at zero momentum, we insist that this observed behavior isperfectly consistent with expectations.
The general solution of the equivalent quadratic equation(17) is
๐ =
(โ๐ต + โ๐ต2 โ 4๐ด๐ถ)
2๐ด
โ โ๐ต
2๐ด+ โ๐ถ
1
๐ด+
๐ต2
8๐ดโ๐ด๐ถ1
+ โ โ โ ,
(28)
where we have disregarded the negative root and haveperformed a binomial expansion of (1 + ๐ต2/4๐ด๐ถ
1)
1/2, with๐ถ
1= โ๐ถ. Upon feeding numerical values, we observe that the
expansion parameter (๐ต2/4๐ด๐ถ1) is much smaller than unity
for our entire density variation, which justifies the validity ofthis expansion for our analysis. The same exercise also allowsus to safely disregard higher-order terms and simply write
๐hyp โ โ๐ถ
1
๐ดโ๐ต
2๐ด. (29)
Further, with the same justification as in the nuclear mattercase, both (๐ถ1/๐ด)
1/2 and (1/๐ด) can also be expanded bino-mially. For example, for the second term, this gives
|๐ต|
2๐ด=|๐ต|
2+ O(
๐๐๐๐
๐
๐4
๐ท๐
) , (30)
where the contribution from these higher-order terms issmaller owing to the large denominator, prompting us toretain only the first-order terms. Here, we point out that since๐ต = ยฑ|๐ต| (+ sign for ๐ท+
๐and โ sign for ๐ทโ
๐), this term,
which represents the Weinberg-Tomozawa contribution tothe dispersion relations, breaks the degeneracy of the ๐ท
๐
mesons. On the other hand, the first term in (29) is commonbetween ๐ท+
๐and ๐ทโ
๐mesons. Thus, the general solution of
the ๐ท๐meson dispersion relations in the hyperonic matter
context can be written as
๐hyp = ๐com โ ๐brk, (31)
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where
๐com = โ๐ถ1
๐ดโ ๐
๐ท๐
[1 โ (๐1
4๐2
๐ท๐
โ
(๐+๐ป)
๐๐
๐)
โ (๐
2
4๐2
๐ท๐
(2โ
ฮ
๐๐
๐+ โ
(๐ปโฮ)
๐๐
๐)) + (
(๐+ ๐
๐)
2โ2๐๐ท๐
)] ,
(32)
๐brk =|๐ต|
2๐ดโ|๐ต|
2= [
1
4๐2
๐ท๐
(2โ
ฮ
๐๐ + โ
(๐ปโฮ)
๐๐)] . (33)
Thus, ๐๐ท+
๐
= ๐com โ ๐brk and ๐๐ทโ๐
= ๐com + ๐brk(where ๐brk is necessarily positive), which readily explainswhy the mass of๐ท+
๐drops more than that of๐ทโ
๐, with density.
Also, this formulation readily accounts for symmetric fallof the medium masses of ๐ท+
๐and ๐ทโ
๐mesons, about ๐com
in Figure 3. Trivially, it may also be deduced that ๐brk isextinguished in nuclear matter, so that the mass degeneracyof ๐ท+
๐and ๐ทโ
๐is recovered from these equations. However,
we observe that though the curve corresponding to ๐com isexactly bisecting the masses of ๐ท+
๐and ๐ทโ
๐at small densities,
this bisection is no longer perfect at high densities. This canbe understood in the following manner. In essence, we arecomparing the density dependence of a function ๐(๐๐
๐), with
the functions ๐(๐๐ ๐) ยฑ ๐(๐
๐). Since scalar densities fall sublin-
early with the number density, it follows that the fall can notbe absolutely symmetric at any arbitrary density. In fact, sincea decreasing function of scalar density will fall faster than thatof a number density, one expects ๐com to lean away from thecurve for๐ทโ
๐meson and towards the curve corresponding to
๐ท+
๐, which is exactly what is observed in Figure 3.Since this disparity between ๐ท+
๐and ๐ทโ
๐originates from
the Weinberg-Tomozawa term, whose magnitude is directlyproportional to the hyperonic number densities, one expectsthis disparity to grow with an increase in the strangenesscontent of the medium.The same also follows from the aboveformulation, since the mass splitting between the two, ฮ๐ =(๐๐ทโ
๐
โ ๐๐ท+๐
) โก (๐๐ทโ๐
โ ๐๐ท+๐
) = โ2๐brk, should grow withboth ๐
๐ at fixed ๐
๐ต(i.e., a larger proportion of hyperons) and
๐๐ตat fixed (nonzero) ๐
๐ (i.e., a larger hyperonic density), in
accordancewith (33). Naively, one expects this๐๐ dependence
to be shared by the two ๐ท๐mesons; however, closer analysis
reveals that this is not the case, as shown in Figure 4 where weconsider the ๐
๐ dependence of the ๐ท
๐meson medium mass.
It is observed that while the ๐๐ dependence for ๐ท+
๐meson is
quite pronounced, the same for the๐ทโ๐is conspicuously sub-
dued. Counterintuitive as it may apparently be, this observedbehavior follows from the above formulation. Since ๐๐ทโ
๐
=
๐com +๐brk, at any fixed density, the first of these is a decreas-ing function of ๐
๐ , while the second increases with ๐
๐ . These
opposite tendencies are responsible for the weak ๐๐ depen-
dence of๐ทโ๐meson.On the other hand, because๐
๐ท+
๐
= ๐comโ
๐brk, the two effects add up to produce a heightened overalldecreasewith๐
๐ for the๐ท+
๐meson, aswe observe in the figure.
A comparison of these hyperonic matter (๐๐ ฬธ= 0) solu-
tions with the nuclear matter (๐๐ = 0) solutions is shown in
Figure 5 for typical values of the parameters. It is observed
0 1 2 3 4 5 61800
1850
1900
1950
2000
(๐B/๐0)
DโSD+S
๐com
Ener
gy๐(k=0)
(MeV
)Figure 3:๐ท
๐mesonmass degeneracy breaking in hyperonic matter,
as a function of baryonic density, for typical values of the otherparameters (๐ = 100MeV, ๐
๐ = 0.5, and ๐ = 0.5). The medium
masses of ๐ท+๐and ๐ทโ
๐are observed to fall symmetrically about ๐com
(see text).
from Figure 5(a) that, at small densities, the nuclear mattercurve, which represents both ๐ท+
๐and ๐ทโ
๐, bisects the mass
degeneracy breaking curve, akin to ๐com in Figure 3. How-ever, at larger densities, both of these ๐
๐ ฬธ= 0 curves drop
further than the ๐๐ = 0 curve. From a casual comparison of
expressions, it appears that the nuclear matter solution, givenby (27), also enters the hyperonic matter solution, (32), albeitas a subset of ๐com. It is tempting to rearrange the latter then,such that this nuclear matter part is separated out, generatingan expression of the type ๐hyp = ๐nucl + ๐(๐๐, ๐
๐
๐)|
๐=๐ป, which
would also entail the segregation of the entire ๐๐ dependence.
However, careful analysis reveals that it is impossible toachieve this kind of a relation, since, in spite of an identicalexpression, in hyperonic matter, the RHS of (27) does notgive the nuclear matter solution. This is because the nuclearmatter solution involves the scalar fields (and scalar densitiescomputed using these scalar fields) obtained as a solution ofcoupled equations in the ๐๐ = 0 situation, which are radicallydifferent from the solutions obtained in the ๐๐ ฬธ= 0 case. (Thishas been observed to be the case, in almost every treatmentof hyperonic matter within this effective model but perhapsmost explicitly in [46].)Thus, one can not retrieve the nuclearmatter solution from the scalar fields obtained as solutionsin the ๐๐ ฬธ= 0 case; moreover, due to the complexity of theconcerned equations, it is impossible to achieve a closed-formrelation between the values of the scalar fields (and hence alsothe scalar densities) obtained in the two cases. In fact, thelow-density similarity between Figures 3 and 5(a) might leadone to presume that ๐com reduces to ๐nucl at small densities.This expectation is fueled further by their comparison in
-
Advances in High Energy Physics 11
0 1 2 3 4 5 61800
1850
1900
1950
2000
(๐B/๐0)
fs = 0
fs = 0.3fs = 0.5
Ener
gy๐(k=0)
(MeV
)
(a) ๐ท+๐
0 1 2 3 4 5 61800
1850
1900
1950
2000
(๐B/๐0)
fs = 0
fs = 0.3fs = 0.5
Ener
gy๐(k=0)
(MeV
)
(b) ๐ทโ๐
Figure 4: The sensitivity of the medium mass of (a) ๐ท+๐and (b) ๐ทโ
๐mesons, to the strangeness fraction ๐
๐ , at typical values of temperature
(๐ = 100MeV) and isospin asymmetry parameter (๐ = 0.5).
0 1 2 3 4 5 61800
1850
1900
1950
2000
(๐B/๐0)
DโS (fs = 0.5)
D+S (fs = 0.5)
Both (fs = 0)
Ener
gy๐(k=0)
(MeV
)
(a)
0 1 2 3 4 5 61800
1850
1900
1950
2000
(๐B/๐0)
๐com
๐Nucl
Ener
gy๐(k=0)
(MeV
)
(b)
Figure 5: (a) A comparison of medium mass of ๐ท๐mesons, in nuclear and hyperonic matter, maintaining the same values of parameters as
before. (b) Comparison of the nuclear matter solution, ๐nucl, with the ๐com in hyperonic matter, as observed in Figure 3.
-
12 Advances in High Energy Physics
0
0 250 500 750 1000
๐B = ๐0๐B = 4๐0
U(๐,k)
(MeV
)
โ50
โ100
โ150
โ200
Momentum, k (MeV)
(a) ๐ท+๐
0 250 500 750 1000
๐B = ๐0๐B = 4๐0
U(๐,k)
(MeV
)
0
โ50
โ100
โ150
โ200
Momentum, k (MeV)
(b) ๐ทโ๐
Figure 6:The optical potentials of the (a)๐ท+๐and (b)๐ทโ
๐mesons, as a function of momentum ๐ (โก|๏ฟฝโ๏ฟฝ|), in cold (๐ = 0), asymmetric (๐ = 0.5)
matter, at different values of ๐๐ต. In each case, the hyperonic matter situation (๐
๐ = 0.5), as described in the legend, is also compared against
the nuclear matter (๐๐ = 0) situation, represented by dotted lines.
Figure 5(b), which shows clearly that they coincide at smalldensities. However, in light of the above argument, it followsthat they are absolutely unrelated. Their coincidence at smalldensities can be explained by invoking the relation ๐๐
๐โ ๐๐ in
this regime, using which (32) reduces to
๐com โ ๐๐ท๐
(1 +
(๐+ ๐
๐)
2โ2๐๐ท๐
โ๐ 1
4๐2
๐ท๐
๐๐ต)
โ ๐๐ท๐
(๐ 2ฮ
4๐2
๐ท๐
) ,
(34)
with ๐ 1 = (๐1 + ๐2) โ 1.28๐1, with our choice of parameters,๐ 2 (โก๐2) = 0.22๐ 1, and with the difference term, ฮ = (๐
๐
ฮ0 +
๐๐
ฮโ โ๐
๐
๐โ๐
๐
๐).The contribution from this second term in (34)
is significantly smaller than the first term at small densities,owing to the smaller coefficient, as well as the fact that thisdepends on the difference of densities rather than on theirsum (like the first term). This first part of (34) can be likenedto the approximate nuclear matter solution at small ๐๐ต, as weconcluded from (27). The marginal increase of ๐ above ๐1is compensated by a marginal increase in ๐ for the ๐
๐ ฬธ= 0
case, as compared to the ๐๐ = 0 situation, and hence thetwo curves look approximately the same at small densities inFigure 5(b). At larger densities, however, this simple picturebreaks down, and the fact that they are unrelated becomesevident. Nevertheless, we may conclude in general fromthis comparison that the inclusion of hyperonic degrees offreedom in the hadronic medium makes it more attractive,regarding ๐ท
๐mesonsโ in-medium interactions, especially at
large ๐๐ต. From the point of view of the ๐ท๐ meson dispersionrelations, this is conclusively because of the contributionsfrom the extra, hyperonic terms, which result in an overallincrease of the coefficient ๐ดhyp significantly above ๐ดnucl,particularly at large ๐๐ .
Finally, we consider momentum-dependent effects byinvestigating the behavior of the in-medium optical poten-tials for the ๐ท
๐mesons. These are shown in Figure 6, where
we consider them in the context of both asymmetric nuclear(๐
๐ = 0) and hyperonic matter (๐
๐ = 0.5), as a function of
momentum ๐ (=|๏ฟฝโ๏ฟฝ|) at fixed values of the parameters ๐๐ต, ๐,
and ๐. As with the rest of the investigation, the dependenceof the in-medium optical potentials on the parameters ๐and ๐ is quite weak; hence, we neglect their variation in thiscontext. In order to appreciate the observed behavior of theseoptical potentials, we observe that as per its definition, (21),at zero momentum, optical potential is just the negative ofthe mass drop of the respective meson (i.e., ๐(๐ = 0) =โฮ๐(๐ = 0) โก ฮ๐(๐
๐ต, ๐, ๐, ๐
๐ )). It follows then that, at
๐ = 0, the two ๐ท๐mesons are degenerate in nuclear matter;
moreover, it is observed that this degeneracy extends to thefinite momentum regime.This is because, from (15) and (16),the finite momentum contribution is also common for ๐ท+
๐
and ๐ทโ๐in nuclear matter. In hyperonic matter, however,
nonzero contribution from these terms breaks the degeneracyin the ๐ = 0 limit (as was already discussed before), and finitemomentum preserves this nondegeneracy. Consequently, atany fixed density, the curves for different values of ๐
๐ differ
in terms of their ๐ฆ-intercept but otherwise appear to runparallel. Further, as we had reasoned earlier for the ๐ = 0
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Advances in High Energy Physics 13
case, the effect of increasing ๐๐ on the mass drops of ๐ท+
๐and
๐ทโ
๐is equal and opposite about the nuclear matter situation at
small densities, which readily explains the behavior of opticalpotentials for these mesons at ๐
๐ต= ๐
0. At larger densities,
for example, the ๐๐ต= 4๐
0case shown in Figure 6, the lower
values of optical potentials for both๐ท+๐and๐ทโ
๐, as compared
to the ๐๐ = 0 case, can be immediately reconciled with thebehavior observed in Figure 5. In fact, the large differencebetween the๐ท+
๐optical potentials for the ๐๐ = 0 and ๐๐ = 0.5
cases, as compared to that for ๐ทโ๐, is a by-product of their
zero-momentum behavior, preserved at nonzeromomenta asa consequence of (15) and (16).
We next discuss the possible implications of thesemedium effects. In the present investigation, we haveobserved a reduction in the mass of the ๐ท
๐mesons, with
an increase in baryonic density. This decrease in ๐ท๐meson
mass can result in the opening up of extra reaction channelsof the type ๐ด โ ๐ท+
๐๐ท
โ
๐in the hadronic medium. In fact,
a comparison with the spectrum of known charmoniumstates [47, 65] sheds light on the possible decay channels,as shown in Figure 7. As a starting approximation, we haveneglected the variation of energy levels of these charmoniawith density, since we intuitively expect medium effects tobe larger for ๐ ๐ (or ๐๐ ) system, as compared to ๐๐ system.However, this approximation can be conveniently relaxed,for example, as was done in [46, 66] for charmonia andbottomonia states, respectively. It follows from Figure 7that above certain threshold density, which is given by theintersection of the ๐ท
๐meson curve with the respective
energy level, the mass of the ๐ท+๐๐ท
โ
๐pair is smaller than
that of the excited charmonium state. Hence, this decaychannel opens up above this threshold density. The mostimmediate experimental consequence of this effect wouldbe a decrease in the production yields for these excitedcharmonia in collision experiments. Additionally, since anextra decay mode will lower the lifetime of the state, oneexpects the decay widths of these excited charmonia to bemodified as a result of these extra channels. Though it can bereasoned right away that reduction in lifetime would causea broadening of the resonance curve, more exotic effects canalso result from these level crossings. For example, it has beensuggested in [67] that if the internal structure of the hadronsis also accounted for, there is even a possibility for thesein-medium widths to become narrow. This counterintuitiveresult originates from the node structure of the charmoniumwavefunction, which can even create a possibility for thedecay widths to vanish completely at certain momenta of thedaughter particles. Also, if the medium modifications of thecharmonium states are accounted for, the decay widths willbe modified accordingly (as was observed in the context ofthe๐ทmesons, in [46]).
Apart from this, one expects signatures of these mediumeffects to be reflected in the particle ratio (๐ท+
๐/๐ท
โ
๐). A produc-
tion asymmetry between๐ท+๐and๐ทโ
๐mesons might be antici-
pated owing to their unequal mediummasses, as observed inthe current investigation. Also, due to the semileptonic andleptonic decaymodes of the๐ท๐ mesons [47], one also expectsthat, accompanying an increased production of ๐ท๐ mesonsdue to a lowered mass, there would be an enhancement
0 1 2 3 4 5 63600
3650
3700
3750
3800
3850
3900
3950
fs = 0
fs = 0.2
fs = 0.3
fs = 0.5
X(3940)
๐c2(2P) (3927)
๐(3770)
X(3915)
X(3872)
๐C(2S) (3639)
(๐B/๐0)
๐(2S) (3686)
Ener
gy๐(k=0)
forD
+ SDโ S
pair
(MeV
)
Figure 7: Mass of the ๐ท+๐๐ท
โ
๐pair, compared against the (vacuum)
masses of the excited charmonia states, at typical values of temper-ature (๐ = 100MeV) and asymmetry (๐ = 0.5), for both nuclearmatter (๐
๐ = 0) and hyperonic matter (๐
๐ = 0.2, 0.3, and 0.5)
situations. The respective threshold density (see text) is given by thepoint of intersection of the two concerned curves.
in dilepton production as well. Further, we expect thesemedium modifications of the ๐ท
๐mesons to be reflected in
the observed dilepton spectra, due to the well-known factthat dileptons, with their small interaction cross section inhadronic matter, can serve as probes of medium effects incollision experiments [68, 69]. The attractive interaction ofthe๐ท
๐mesons with the hadronic medium might also lead to
the possibility of formation of๐ท๐-nucleus bound state, which
can be explored at the CBM experiment at FAIR in the futurefacility at GSI [70].
We now discuss how the results of the present investi-gation compare with the available literature on the mediumeffects for pseudoscalar ๐ท
๐(1968.5) mesons. As has already
been mentioned, [42โ45] have treated the medium behaviorof ๐ท๐ mesons, using the coupled channel framework. Thebroad perspective that emerges from all these analyses is thatof an attractive interaction in the medium [44, 45] between๐ท๐ mesons and baryon species like the nucleons or theฮ, ฮฃ, ฮhyperons [42โ45], which is consistent with what we haveobserved in this work. Additionally, one observes in theseapproaches an attractive medium interaction even with thecharmed baryons [43]. Some of these interaction channelsalso feature resonances, which is reflected in the relevantscattering amplitudes picking up imaginary parts as well, forexample, the ๐
๐๐ (2892) state in the ๐ท+
๐๐ channel [42, 43].
Treating such resonances comes naturally to the coupled
-
14 Advances in High Energy Physics
channel framework, since, by default, the strategy is aimedat considering the scattering of various hadron species offone another and assessing scattering amplitudes and crosssections, and so forth [10, 11]. In the present work, we haveinvestigated the in-medium masses of the๐ท
๐mesons arising
due to their interactions with the baryons and scalar mesonsin the nuclear (hyperonic) medium and have not studied thedecays of these mesons. As has already been mentioned themass modifications of the ๐ท๐ mesons can lead to openingup of the new channels for the charmonium states decayingto ๐ท+
๐๐ท
โ
๐at certain densities as can be seen from Figure 7.
In fact, even if one contemplates working along the linesof [46] for calculating these decay widths, the situation is abit more complicated for these ๐ท๐ mesons, since the states๐(3872),๐(3915), and๐(3940), which are highly relevant inthis context (as one can see from Figure 7), are not classifiedas having clear, definite quantum numbers in the spectrum ofexcited charmonium states [47]. This makes the assessmentof medium effects for these states more difficult in com-parison, since the identification of the relevant charmoniumstates with definite quantum numbers in the charmoniumspectrum, such as 1๐ for ๐ฝ/๐, 2๐ for ๐(3686), and 1๐ท for๐(3770), was a crucial requirement for evaluating their massshifts in the medium in the QCD second-order Stark effectstudy in [46]. For some of these unconventional states, forexample,๐(3872), this absence of definite identification withstates in the spectrum has led to alternative possibilitiesbeing explored for these states. A prominent example is thepossibility of amolecular structure for this๐(3872) state [65],borne out of contributions from the (๐ท+๐ทโโ + ๐ทโ๐ทโ+) and(๐ท
0๐ท
โ0
+ ๐ท0
๐ทโ0) components in the ๐ -wave.
Lastly, we point out that the medium effects describedin this paper, and the possible experimental consequencesentailed by these, are especially interesting in wake of theupcoming CBM [70] experiment at FAIR, GSI, where highbaryonic densities are expected to be reached. Due to thestrong density dependence of thesemediumeffects, we expectthat each of these mentioned experimental consequenceswould intensify at higher densities and should be palpable inthe aforementioned future experiment.
5. Summary
To summarize, we have explored the properties of๐ท๐mesons
in a hot and dense hadronic environment, within the frame-work of the chiral ๐๐(3)
๐ฟร ๐๐(3)
๐ model, generalized to
๐๐(4). The generalization of chiral ๐๐(3) model to ๐๐(4)is done in order to derive the interactions of the charmedmesons with the light hadrons, needed for the study of thein-medium properties of the๐ท๐ mesons in the present work.However, realizing that the chiral symmetry is badly brokenfor the ๐๐(4) case due to the large mass of the charm quark,we use the interactions derived from ๐๐(4) for the๐ท๐ mesonbut use the observed masses of these heavy pseudoscalarmesons as well as empirical/observed values of their decayconstants [30]. The๐ท
๐mesons have been considered in both
(symmetric and asymmetric) nuclear and hyperonic matter,at finite densities that extend slightly beyond what can beachieved with the existing and known future facilities and
at both zero and finite temperatures. Due to net attractiveinteractions in the medium, ๐ท
๐mesons are observed to
undergo a drop in their effective mass. These mass drops arefound to intensify with an increase in the baryonic densityof the medium, while being largely insensitive to changesin temperature as well as the isospin asymmetry parameter.However, upon adding hyperonic degrees of freedom, themass degeneracy of ๐ท+
๐and ๐ทโ
๐is observed to be broken.
The mass splitting between ๐ท+๐and ๐ทโ
๐is found to grow
significantly with an increase in baryonic density as well asthe strangeness content of the medium. Through a detailedanalysis of the in-medium dispersion relations for the ๐ท๐mesons, we have shown that the observed behavior followsprecisely from the interplay of contributions from variousinteraction terms in the Lagrangian density. We have brieflydiscussed the possible experimental consequences of thesemedium effects, for example, in the ๐ท+
๐/๐ท
โ
๐ratio, dilepton
spectra, possibility of formation of exotic ๐ท๐-nucleus bound
states, and modifications of the decays of charmonium states๐ท
+
๐๐ท
โ
๐pair in the hadronic medium. The medium modifi-
cations of the ๐ท๐mesons are expected to be considerably
enhanced at large densities and hence the experimentalconsequences may be accessible in the upcoming CBMexperiment, at the future facilities of FAIR, GSI [70].
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
Divakar Pathak acknowledges financial support from Uni-versity Grants Commission, India (Sr. no. 2121051124, Ref.no. 19-12/2010(i)EU-IV). Amruta Mishra would like to thankDepartment of Science andTechnology,Government of India(Project no. SR/S2/HEP-031/2010), for financial support.
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